I paint on dye using a paint brush. By using only dye the discs remain tournament legal. If regular paint were used it would add thickness to the disc and you would no longer be able to use the disc at PDGA tournaments.

Very cool idea dying your kids hands! I love it.
Tape doesn't work well. Get vinyl if you can. If you can't get vinyl, plain contact paper works pretty well. I still mask the edges with contact paper so I can get away with using less vinyl on each disc.

here's my most recent dye and i think my best so far. the orange soaked through my mask and still a little color bleed but i'm real happy with. it's a star Ontario Roc. I also doctored it a little with a sharpie (second vs first photo).